Question

Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. if it is not, enter none. f(x, y) = (2x − 4y) i + (−4x + 10y − 9) j

Answer 1

$\mathbf f(x,y)=(2x-4y)\,\mathbf i+(-4x+10y-9)\,\mathbf j$

is conservative if

$\dfrac{\partial(2x-4y)}{\partial y}=\dfrac{\partial(-4x+10y-9)}{\partial x}$

It's clear that this is true, since both partial derivatives reduce to -4, so $\mathbf f$ is indeed a conservative vector field.

Since $\mathbf f(x,y)$ is conservative, there must be some scalar function $f(x,y)$ such that

$\nabla f(x,y)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j=\mathbf f(x,y)$
$\implies\dfrac{\partial f}{\partial x}=2x-4y$

Integrating with respect to $x$ yields

$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=f(x,y)=x^2-4xy+g(y)$

and differentiating with respect to $y$ yields

$\dfrac{\partial f}{\partial y}=-4x+g'(y)=-4x+10y-9$
$\implies g'(y)=10y-9$
$\implies g(y)=\displaystyle\int g'(y)\,\mathrm dy=5y^2-9y+C$

Thus the scalar potential of $\mathbf f(x,y)$ is

$f(x,y)=x^2-4xy+5y^2-9y+C$

where $C$ is an arbitrary constant.